The Second Derivative
Definition and Notation
The second derivative is a measure of how the rate of change of a function is itself changing, obtained by differentiating a function twice. There are two common notations for the second derivative:
- Leibniz notation: $\frac{d^2y}{dx^2}$
- Lagrange notation: $f''(x)$
For example, if we have a function $f(x) = x^3$, its first derivative is $f'(x) = 3x^2$, and its second derivative is $f''(x) = 6x$.
Graphical Interpretation
The second derivative provides crucial information about the shape and behavior of a function's graph:
- Concavity: The sign of the second derivative determines the concavity of the function.
- if $f''(x) > 0$, the function is concave up (shaped like a cup).
- If $f''(x)< 0$, the function is concave down (shaped like an inverted cup).
- Inflection Points: Points where the concavity changes (i.e., where $f''(x) = 0$ or is undefined) are called inflection points. Because $\frac{dy}{dx}$ is at a maximum or minimum at this point, the tangent line will change from increasing to decreasing—concave down—or change from decreasing to increasing—concave up —hence why they are studied.
Consider the function $f(x) = x^3$:
- $f'(x) = 3x^2$
- $f''(x) = 6x$
The function is concave up when $x > 0$ and concave down when $x
< 0$. The inflection point occurs at $x = 0$.
